Compatibility summary
You will need a travel adapter in Luxembourg; plug types used there differ from Australia.
Plug comparison
Australia plug types: I
Luxembourg plug types: C F
There is no common plug type between Australia (I) and Luxembourg (C F). You need a travel adapter.
Voltage comparison
Australia: 230V, 50Hz
Luxembourg: 230V, 50Hz
Voltage difference is within a range many devices tolerate; check your appliance label.